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Simplifying 25(0.25 + -1r + r2) = 47r2 (0.25 * 25 + -1r * 25 + r2 * 25) = 47r2 (6.25 + -25r + 25r2) = 47r2 Solving 6.25 + -25r + 25r2 = 47r2 Solving for variable 'r'. Combine like terms: 25r2 + -47r2 = -22r2 6.25 + -25r + -22r2 = 47r2 + -47r2 Combine like terms: 47r2 + -47r2 = 0 6.25 + -25r + -22r2 = 0 Begin completing the square. Divide all terms by -22 the coefficient of the squared term: Divide each side by '-22'. -0.2840909091 + 1.136363636r + r2 = 0 Move the constant term to the right: Add '0.2840909091' to each side of the equation. -0.2840909091 + 1.136363636r + 0.2840909091 + r2 = 0 + 0.2840909091 Reorder the terms: -0.2840909091 + 0.2840909091 + 1.136363636r + r2 = 0 + 0.2840909091 Combine like terms: -0.2840909091 + 0.2840909091 = 0.0000000000 0.0000000000 + 1.136363636r + r2 = 0 + 0.2840909091 1.136363636r + r2 = 0 + 0.2840909091 Combine like terms: 0 + 0.2840909091 = 0.2840909091 1.136363636r + r2 = 0.2840909091 The r term is 1.136363636r. Take half its coefficient (0.568181818). Square it (0.3228305783) and add it to both sides. Add '0.3228305783' to each side of the equation. 1.136363636r + 0.3228305783 + r2 = 0.2840909091 + 0.3228305783 Reorder the terms: 0.3228305783 + 1.136363636r + r2 = 0.2840909091 + 0.3228305783 Combine like terms: 0.2840909091 + 0.3228305783 = 0.6069214874 0.3228305783 + 1.136363636r + r2 = 0.6069214874 Factor a perfect square on the left side: (r + 0.568181818)(r + 0.568181818) = 0.6069214874 Calculate the square root of the right side: 0.779051659 Break this problem into two subproblems by setting (r + 0.568181818) equal to 0.779051659 and -0.779051659.Subproblem 1
r + 0.568181818 = 0.779051659 Simplifying r + 0.568181818 = 0.779051659 Reorder the terms: 0.568181818 + r = 0.779051659 Solving 0.568181818 + r = 0.779051659 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.568181818' to each side of the equation. 0.568181818 + -0.568181818 + r = 0.779051659 + -0.568181818 Combine like terms: 0.568181818 + -0.568181818 = 0.000000000 0.000000000 + r = 0.779051659 + -0.568181818 r = 0.779051659 + -0.568181818 Combine like terms: 0.779051659 + -0.568181818 = 0.210869841 r = 0.210869841 Simplifying r = 0.210869841Subproblem 2
r + 0.568181818 = -0.779051659 Simplifying r + 0.568181818 = -0.779051659 Reorder the terms: 0.568181818 + r = -0.779051659 Solving 0.568181818 + r = -0.779051659 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.568181818' to each side of the equation. 0.568181818 + -0.568181818 + r = -0.779051659 + -0.568181818 Combine like terms: 0.568181818 + -0.568181818 = 0.000000000 0.000000000 + r = -0.779051659 + -0.568181818 r = -0.779051659 + -0.568181818 Combine like terms: -0.779051659 + -0.568181818 = -1.347233477 r = -1.347233477 Simplifying r = -1.347233477Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.210869841, -1.347233477}
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